I want to rotate a parent object and see how far a certain vertex of the child object has moved
You can get this information with a vertex parent, and a driver.

Start by adding two empties at the corner of your cube, the point you want to measure, (I named my empties "corner" and "original location").
Vertex parent one of the empties to the corner vertex of the cube. (Select the empty, select cube Tab into edit mode, select the vertex then press CtrlP)


Now you know the original location of that vertex and the vertex parented empty will tell you the new location.

Add another object, any object (does not matter at all). Put a single driver on one of its transform channels.


Now open up the Graph Editor switch to the Drivers mode and click on the transform channel on the left. When you open up the proprieties region N, you will see something similar to the image above.

You first want you make sure the variable name is in the expression field. Now set the variable to Distance, and put in the two empties you added earlier as the objects in that variable.
The output of that driver is the distance the corner of your cube has moved. If you have your scene units set to Meters, then it will be in millimeters.